Tracy Zager

Breaking the Cycle

The majority of elementary school teachers had negative experiences as math students, and many continue to dislike or avoid mathematics as adults. We’ll look at how we can better understand and support our colleagues, so they can reframe their personal relationships with math and teach better than they were taught.

Call to Action

  • This is a collection of words mathematicians’ use when they talk about mathematics. Discuss it with your colleagues, making an extra effort to include everyone.
  • Each person should choose a word that appeals most to him or her. It should be a word that’s not currently a big part of your math teaching and learning, but you wish it were.
  • Using your colleagues as resources and collaborators, make the word you chose central to the planning, teaching, and learning of your next lesson. Don’t skimp on the conversation with your team–that’s part of the point!
  • Teach the lesson. Afterwards, get back together with your colleagues and talk about it. What was different for each of you? What was different for your students?
  • If it works for you, consider sharing the image and exploring it with your students as well. There are lots of possibilities here.
  • Write a few paragraphs here so we can learn together. Describe what happened and what you learned. What ideas do you and your colleagues have for building on this exercise?

About the Speaker

Tracy is a fourth-grade teacher at heart. When her daughters came along, she gave up her own classroom to work with pre-service teachers and their in-service mentor teachers. After several years in adult education, she began field research for her upcoming book, Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms (to be published in 2016 by Stenhouse Publishers). She also works with schools as a coach/consultant, and loves learning together with teachers and students over time.

Updated 2015 Apr 21: Livetweeting

Check out the collection on Storify.

41 thoughts on “Tracy Zager”

  1. I thoroughly enjoy any time that I can have with Tracy to talk about the teaching and learning of math. She’s a wonderful resource.

    1. Dylan,
      This is an awesome, thoughtful response. I’ve been mulling it over for a few days and will keep doing so. A few thoughts:

      First off, I really like novice/expert research myself. One of my all time favorite articles is this one from Sam Wineburg, in the context of thinking historically: Thought you might enjoy it. I always find it delightful, and not intimidating.

      The other article that came to mind was Skemp, on the difference between relational and instrumental understanding of math:

      Neither of those will help you reconcile what you’re struggling with, but I bet they will both give you food for thought.

      My main reaction was to worry a little less than you are about the impact of novice-ness on my Call to Action. Reading Willingham’s paper should make me worry, but my experience with my daughters counters it. They are now 8 and almost 6, and what I’ve observed is that they don’t need expertise to play, wonder, discover, imagine, ask, delight, etc. That’s the part that comes naturally. Not-knowing doesn’t slow them down a bit. True, they don’t think like experts in that they don’t have a framework for connections, or a context for relationships. But I find they have this fearless tendancy to wonder that hasn’t been taught out of them…yet.

      There was a segment of my Shadow Con talk that I had to cut for time purposes. If I did it correctly, the graph is attached. The data come from McAnallen, R. 2010. Examining mathematics anxiety in elementary classroom teachers (Doctoral dissertation, University of Connecticut). She asked teachers with math anxiety: when did your math anxiety start? This graph shows their answers. There’s a lot in this graph that’s interesting, but one of my favorite aspects is that it starts with kindergarten for a reason. In all the studies I’ve ever read about math anxiety, nobody has ever suggested anyone is born with it. I’ve hung out with a lot of 4-year olds, and they all like to count to the biggest number they can think of, find the longest string bean on the plate, pour sand from one container to another and wonder if it will fit or overflow, etc. In other words, all little kids think like mathematicians.

      By first or second grade, most kids I know think math is something that is done in school, on worksheets, and to be good at math means to answer the teacher’s questions right, fast, and easily. They get our societal messages really fast.

      So, at some level, what I want to do for teachers and kids is to reconnect them to the 4-year olds inside them. The ones who saw math as a way to ask and answer their own questions. The ones who saw math in the world around them, and wondered about it, and thought it was fun to figure stuff out.

      In many ways, the older the students, the more difficult it is to uncover that curious little kid inside. But that doesn’t mean we shouldn’t try, right?

      I can’t wait to hear how it goes. Please keep talking with me!

    2. I think the expert/novice research gives us extra tools and sensitivities for helping novices have the kinds of experiences that lead to “expert” feelings for math. Each of the differences between novices and experts links with a strategy we can teach, model, and cue. For example, experts have stored information in long term memory while novices have to hold it in working memory. So, we get novices to write down information to, functionally increasing their working memory.

      Experts see deeper properties, while novices are operating on the superficial. To handle this, we can use notice and wonder discussions where students work together to push beyond the surface level.

      Experts use structures and models, so we help novices build structures and models.

      Finally, it is important to recognize the special contribution that non-experts can often make more easily than the experts: asking great questions.

    1. Thanks for the graphic! I have also recently read/heard(?) that the age when student interest in mathematics sharply decreases – particularly in girls – is in this 3rd-5th grade range, just when math classrooms become more focused on procedures. As a math specialist for grades 3 – 6, I feel that this information alone provides a call to action. We need to change something about what we are doing at these grade levels in particular.

      After hearing your talk, I am wondering if perhaps having teachers reflect on when they enjoyed or didn’t enjoy learning math themselves could give them a new perspective on the math they are teaching in their classroom. Is the math they teach the sort of math they enjoyed in school? Or is it the type of math that turned them off in school? Is it the sort of math that makes them happy to be a math teacher? Or is it the type of math that makes them uncomfortable being a math teacher? Then our question to work on together can become: How can we make more of the math you teach the sort of math you would have liked to do and precisely the sort of math you feel inspired to share with kids?

  2. Alison,
    Great thoughts. I wonder where a conversation around your questions would go with your grade 3 – 6 team members?

    And then I have another question, which is what if the things they liked aren’t very math-y? For example, sometimes people liked math because they liked winning Around the World. So, if one of your teachers answered that way, what would come next?

    That’s part of why I spend a lot of time introducing teachers to math as it really is, as defined by mathematicians: because most of us never experienced math as creative, or intuitive, or beautiful, or satisfying, or worth debating.

    So, while I love the thought of having teachers think about what they enjoyed, I also think part of our job is to make sure they are moving toward more authentic math teaching and learning. Sometimes those will overlap, and sometimes they won’t. That’s where your role as an instructional specialist is invaluable! You can help guide teachers toward teaching math they love, in ways that work well for kids, and in the spirit of the discipline. I’m glad they have you!

    1. I am the Math Intervention (k-5) teacher at our school, as well as the PLC lead teacher. I love the idea of showing the teachers the two wordles and getting them to work towards the positive one!
      I think I might ask our teachers what they liked/disliked about math in their elementary years and see where that conversations goes, too. Like Tracy mentioned, maybe it was things that really aren’t “Math-y”. I still see teachers playing Around the World, and not having deep math talks with students. I know there can be a place for games like that, but I also am learning as I go on in this field, that math talks (number talks as some would say) are SO important!

      We actually did just buy a program, like you talked about in your ShadowCon presentation….a BEAUTIFUL math program called Math Expressions. Embedded within the curriculum and lesson plans are guides to help teachers get that math talk going. I have seen teachers struggling with this, though. Now I am wondering, if that struggle is related to their personal math experiences or how they see their skills & strategies.

      I definitely want to pursue this further with our elementary teachers. Any ideas how to start?

  3. Tracy,
    Thank you for an amazing ShadowMathCon talk, “Breaking the Cycle”. It was compelling and resonated with me from the get-go. Your wordles of how a mathematician views math versus how many pre-service (math-reticent) elementary teachers view math is compelling. You spoke with compassion for those teaching math that carry these feelings about math. If we do not break the cycle and get those teaching math to change how they see math, we are bound to turn out more learners who perpetuate the view of math as boring, rote, sometimes humiliating, etc. rather than beautiful, elegant, playful and the like. I appreciated the few moments I spoke with you after your talk gaining some more insight into why this is such a passionate topic for you.

    Children often come into the system seeing math more as mathematicians do and over time many lose that vision of mathematics and problem solving. It kills me to see this! I believe, as Robert Moses does, that mathematics is a civil right and a key to equality. When I work with teachers, para-educators, parents, and/or administrators I start with that, and our district vision as a compelling “Why” we might want mathematics instruction and education to look differently than it might have for you or me, or even for our students. Now, I’ve added to it pictures of each of the wordles you showed during your talk and ask them to think about which they want for their students/children, and which will allow students to understand and solve the problems which currently face our world as well as those we can’t even imagine. I let the wordles sit on the screen for a few extra seconds and this often elicits great conversation and really gets them thinking about how they might want to change math for their students. They are choosing words now that they want to intentionally plan for in their lessons. I share those wordles throughout professional learning sessions, or bring them up in conversation when working with a group of teachers or individually to use as a benchmark for the work we are doing and will implement in our classrooms as a way to ask, “Will _______ lead to this (mathematician’s wordle) or this (pre-service teacher’s wordle)?

    It is a bit of a cultural shift for many teachers not to be focused on what I refer to as an “answer-focused pedagogy.” But to really open up lessons to allow students to show the depth and breadth of their thinking and allow teachers to focus not only on the answer but looking at students’ understanding to guide their work. It’s been delightful to share this with teachers because they are really making changes and seeing changes in their students. By trying some new things, teachers note that students not normally participating are finding entry points and the teacher is able to better support those students without the typical small group remediation, or other such intervention. They are finding teaching mathematics more enjoyable. It’s like a snowball traveling down a hill: gathering size, speed and momentum as it journeys.

    Thank you for adding your imprint to my own mathematical journey!


    1. Jana,
      Thank you so much for this thoughtful comment! And, of course, I’m over the moon that you’ve found the pair of wordles clarifying and helpful in your work with teachers. I particularly loved this part:

      “They are choosing words now that they want to intentionally plan for in their lessons. I share those wordles throughout professional learning sessions, or bring them up in conversation when working with a group of teachers or individually to use as a benchmark for the work we are doing and will implement in our classrooms as a way to ask, “Will _______ lead to this (mathematician’s wordle) or this (pre-service teacher’s wordle)?”

      That’s such a great use of them! What a great guiding question. I’m so glad you posted it here, so other coaches can try it out. You’ve definitely made my initial call to action better and richer!

      I imagine teachers are finding it incredibly rewarding that a wider range of students are able to join in the mathematical action of the class. It doesn’t have to be the way it usually is–with just a few regulars engaged and the rest of the class checked out. How fantastic for them to see that firsthand. I wonder how they’ll reflect on math teaching over the summer?

      One request. Can you keep me posted on which words you find resonate most with teachers, and they find resonate most with their students? I’m curious about that.

      Thanks again! I’m excited and moved to hear about your work.

  4. I loved this, Tracy. You are right — we need to invest in our teachers and try to break the cycle.

    1. Hi Mike,
      I loved this post, thanks! And I completely agree with this point:

      “If you want to move people from first list of words to the second list of words, a great way to do that is show some of the fun things in math that inspired that second list of words. Looking around, you may be surprised how many of these seemingly high level / abstract math ideas are not only accessible to kids, but are things that just blow kids away.”

      I’ve learned a lot about the possibilities here from reading your blog and watching your videos. Thanks!


  5. Tracy,

    I used your video today with my colleagues. It was tremendously well received! We will likely use it in other contexts which is very exciting. There was a question. You mention the author of a book you are reading. We couldn’t understand clearly the author’s name. Could you please send that along? Thanks so much.

    Maryann Wickett

    1. Hi Maryann!
      Thanks for venturing over here to tell us how it went. I would love to hear more about the other contexts and ideas your team came up with.

      As for the book, I’m trying to think which part of the talk. Maybe when I talked about how students were “oriented toward one another?” That comes from Elham Kazemi and Allison Hintz’s Intentional Talk, from Stenhouse. If that’s not it, let me know the context and I’ll be sure to give you the title. Thanks!

  6. This is a great idea. Even though I wasn’t able to attend the NCTM Conference, I really enjoyed listening to the copy of your talk.

    I totally agree with you that we need to change the cycle. I also work with Elementary Teachers and I have one colleague who feels that “coddling” them helps them feel better about their math teaching abilities. Actually expecting them to work together just like I would expect them to have their students work together does much more to build their confidence.

    I definitely will get your book and I am excited to know when and where you will be doing more presentations and/or discussions.


    1. Kristen,
      That’s really interesting. What would your colleague say we tend to do that “coddles” teachers?

  7. Hello again, Tracy,

    I first saw this video four days ago and responded to you then. Since then I have shared it with my team, with former colleagues, and with the women I work for. The response has been amazing and very important in guiding our current work, developing PD for teachers of grades 3-5 that necessitates that teachers be thinkers about mathematics. We envision our work as a thinking curriculum. We expect and want students to be thinkers. Without thinking teachers, we cannot have student thinkers. What you describe is the journey many teachers must go on: from discomfort with mathematics to being an inspired, thinking math teacher. To do this, there are several things that must be put into place. First, we need to acknowledge the elephant in the room, our discomfort with mathematics. Second, teachers must develop their confidence and belief in themselves as powerful, knowledgable teachers. To do this, they must deepen their own understanding and develop a set of effective instructional tools to use with their students. They must become careful listeners, giving over certain types of control to students and the fear that can happen when a student explains his/her thinking. Teachers need to let students realize they are learners too and don’t have all the answers and sometimes that lack of clarity and quest for solutions is even more empowering for both student and teacher. So far, we have included these ideas into three areas; 1. developing and building teacher content knowledge, 2. providing instructional tools clearly linked to the mathematical learning, and 3. using 1 and 2 to build confidence. This is not new to us, but your video has brought these points into focus and provides a brilliant, articulate place to start the discussion and the journey with teachers, and the video serves as a reminder to us of the importance of recognizing the psychological, personal relationships between teachers and mathematics. We are hoping to use the video as part of the launch for our work which usually takes place with a school or district over the course of a year. Thank you!

    1. Maryann,
      I couldn’t love this comment more. I’m nodding along as I read, especially in this part:

      “To do this, they must deepen their own understanding and develop a set of effective instructional tools to use with their students. They must become careful listeners, giving over certain types of control to students and the fear that can happen when a student explains his/her thinking. Teachers need to let students realize they are learners too and don’t have all the answers and sometimes that lack of clarity and quest for solutions is even more empowering for both student and teacher.”

      Yes, yes, yes! I also love that you’re talking about all of this work in terms of the course of a year. That’s how change really happens: over the course of a year, many times over.

      One thing you might want to have teachers do is create their own wordles or images. Mine were extremes on the continuum. Each teacher’s story is unique, though, with some mix of positive and negative memories and associations. You might consider pausing the video after the two wordles and giving colleagues a piece of paper with nothing but the word “math” printed in the middle.

      Here is an example:

      Teachers can draw, write in prose, make a semantic map, or whatever appeals. They can put a date on it, and tuck it away. Then, at the end of the year, you could give teachers an opportunity to make a new one. I wonder how they might change?

      Just a thought! I do hope you’ll keep me posted all year!

  8. Will keep you posted as our work progresses. Creating Woodles is an excellent suggestion. For a while it was overused, but I haven’t seen it used nearly as often these days. Thanks for the suggestion.

    1. So funny. I mostly hate them because they’ve been overused!

      When I’ve done this exercise with teachers, I didn’t limit them to wordles. Just reflect and write or draw.

  9. Hi Tracy,

    We adopted a new math program this year, which is stretching our teachers in many ways. The program includes a time for number talks, and the lessons are based on inquiry and exploration. The kids are loving it and math time, their conceptual understanding is increasing, and they are loving sharing their thinking. K-2 teachers are doing much better with the program right now than 3-5 teachers. The program assumes that students have been using it since kinder, so the 3-5 students were lacing quite a bit of vocabulary and classroom structure the program builds upon. This is improving, and kids are becoming much more fluent in the activities and learning, however, many teachers are still struggling and feeling very anxious.

    I used the two wordles in a math lead meeting, with K-2 and 3-5 math leaders from each site in our district. I meet with each group once a month and we discuss math, implementation, and other issues that need to be addressed.

    I had them reflect on the teacher vocabulary first, quietly, and then the mathematician vocabulary. I then had the two together with the wording, “Which one are you currently relating to?” I had them discuss this with an elbow partner along with the following thoughts:

    Why do you relate to the one you chose?
    Which one do you think the teachers you work with are relating to?
    Why do you think they are relating to that one?
    How does this affect the atmosphere and learning during math in our classrooms?

    We had a wonderful discussion, with teachers beginning to use words like, “learning community”, “open-minded”, “finding the fun” and generally agreeing that our feelings about what we are teaching will reflect on to the students, and affect their learning.

    It’s a start in a series of discussions about growth mindset, learning along with our students, and creating a learning community that I am planning over the next few months with these groups.

    Thank you so much for your talk, sharing the slides, and your constant encouragement to me to think outside the box.

    1. Wow, Teri, this sounds great! Thank you for telling me about it.

      One thing I’ve done with teachers is to give them a big page that’s blank except for the word “math” written right in the middle. These two wordles I used in the talk are the extremes on a continuum. Every teacher’s story is unique, usually with some mix of positive and negative. So I give them time to draw, write words, write a memory, etc. If I’m new to a community of teachers, I don’t expect them to share out because this stuff is personal and emotional. But often teachers will choose to share. We sometimes will mark that page “Point A.” The idea being that later points are up to us, and that there’s going to be an evolution in feelings and associations about math.

      I’m not saying to go back and do this with this group of teachers. It sounds like they already had a lovely conversation thinking about which one they relate to more. I just wanted to put another strategy in your pocket for another time. 🙂

      Keep me posted as it evolves!

  10. Thanks Tracy,

    I love this idea. Will think about how to incorporate this into other sessions I am having with different sets of teachers. There’s always something going on!


  11. @Robin,
    I can’t seem to reply directly to your comment, so I’m hoping you’ll see this. I’ll shoot you an email to make sure. So, I actually just gave a talk about next steps in Nashville at the Regional NCTM, which I’ll reprise at NCSM in Oakland. Any chance you’ll be there? The short version is:

    1) Talk about the larger contexts. I shared all this data with teachers, and then some. We really worked at it. Then we spent some time exploring what math really is–distinct from school math. We read 4 picture books I love a lot:
    On a Beam of Light by Berne,
    Blockhead by D’Agnese,
    Infinity and Me by Hosford,
    The Boy Who Loved Math by Heiligman.
    We also explored, numberphile,, etc. And we talked a lot about what they noticed in these resources. What does it mean to do mathematics?

    2) Do lots of math together. We used a lot of really accessible, visual tasks from the math twitter blog o sphere to start doing math together. Things like looking at a photograph of a crate of peaches and asking how many and how did you see them? I introduced them to resources like estimation 180. We played Four Strikes and You’re Out. Doing math was essential.

    3) Teach Each Other. I wanted to teach a whole bunch of new instructional routines like Number Talks, Choral Counting, Counting Collections, etc. I knew teachers needed to practice them to get them. And I knew I wanted to create a sustainable PLC where teachers would lean on each other. So, we had a summer institute without kids where I would introduce a strategy, then they would plan it and take turns teaching each other. All kinds of obstacles came down and true collaboration began. It was hard at first, but SO powerful.

    Those are where I’d start. Keep me posted!

    1. Sorry, no chance I will be in Oakland…but thanks for all of the tips! I will check into those books.

  12. Hi Tracy,

    Just wanted to let you know that that your call to action is resonating once again in Ontario, Canada and in French! Following the webcast by Julie Mondoux, my colleague and I decided to do something similar for the teachers of our school board during a PD day. We want teachers to realize that their relationship with math has an important impact to student learning and that our “solution” to overcome those obstacles is to do math together (teachers). In the same way that we ask students to collaborate, communicate their thinking with their peers, question the reasoning of others, we will ask teachers to do the same. We’ll be “broadcasting” on April 11th then we will leave for San Francisco for NCSM/NCTM. If interested, I can let you kow how everything went and what the reaction was from our teachers.

    1. Hi Pierre,
      Thanks so much for letting me know. Amazing. Can’t wait to hear all about it in SF!

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